Sunday, 9 December 2012

Apportionment

[I wrote this in July, but never got round to posting it.]

Last weekend I visited the U.S. Capitol in Washington, D.C., with my family, and I learned that the House of Representatives has 435 seats which are appointed
so that each state has a number of seats that is proportional to its population.
It sounded simple when the tour guide said it, but I wondered how are fractions handled fairly?
Simply rounding off quotas doesn't work—firstly because some states could get no seats, which would be unfair, and
secondly, how do you make sure that the rounding is both fair and assigns all 435 seats?

When I got home I read about the apportionment problem, as it is known, which has a long and interesting history.
Wikipedia [1] is a good read, as usual; and [2] goes into the history and mathematics of different apportionment algorithms in depth, at least one of which suffers causes a paradox.
Here I'm interested in looking at the algorithm that is used today to calculate apportionments for the House of Representatives,
and why it is considered to be the fairest.

The Algorithm

The algorithm in use today for apportioning seats is due to Huntington and Hill and is known as the Huntington-Hill method, or the method of equal proportions.
It's best understood as a dynamic process, which works as follows:

To start, each state is given one seat. (This ensures that states with relatively small populations, like Wyoming, get at least one seat.)
Then, each remaining seat is allocated in turn to the state is allocated to the state with the
highest priority, where the priority of a state of population \(P\) and \(n\) previously-allocated seats is defined as

\begin{align}
\frac {P} {\sqrt{n(n+1)}}\label{pri}
\end{align}

We'll see why the priority is defined as it is below, but for now notice that it is approximately \(P/n\), so the seat is given to the state
that has the least number of representatives per person, roughly speaking.

Results for the 2010 Census

Running the algorithm for the state populations from the 2010 Census (using a program I wrote [5])
gives the following apportionment, which agrees with the U.S. Census Bureau [3]. (The quota column is the percentage of the population for each state.)

State

Seats

Population

Quota

People per representative

Alabama

7

4802982

6.76

686140

Alaska

1

721523

1.02

721523

Arizona

9

6412700

9.02

712522

Arkansas

4

2926229

4.12

731557

California

53

37341989

52.54

704565

Colorado

7

5044930

7.10

720704

Connecticut

5

3581628

5.04

716325

Delaware

1

900877

1.27

900877

Florida

27

18900773

26.59

700028

Georgia

14

9727566

13.69

694826

Hawaii

2

1366862

1.92

683431

Idaho

2

1573499

2.21

786749

Illinois

18

12864380

18.10

714687

Indiana

9

6501582

9.15

722398

Iowa

4

3053787

4.30

763446

Kansas

4

2863813

4.03

715953

Kentucky

6

4350606

6.12

725101

Louisiana

6

4553962

6.41

758993

Maine

2

1333074

1.88

666537

Maryland

8

5789929

8.15

723741

Massachusetts

9

6559644

9.23

728849

Michigan

14

9911626

13.94

707973

Minnesota

8

5314879

7.48

664359

Mississippi

4

2978240

4.19

744560

Missouri

8

6011478

8.46

751434

Montana

1

994416

1.40

994416

Nebraska

3

1831825

2.58

610608

Nevada

4

2709432

3.81

677358

New Hampshire

2

1321445

1.86

660722

New Jersey

12

8807501

12.39

733958

New Mexico

3

2067273

2.91

689091

New York

27

19421055

27.32

719298

North Carolina

13

9565781

13.46

735829

North Dakota

1

675905

0.95

675905

Ohio

16

11568495

16.28

723030

Oklahoma

5

3764882

5.30

752976

Oregon

5

3848606

5.41

769721

Pennsylvania

18

12734905

17.92

707494

Rhode Island

2

1055247

1.48

527623

South Carolina

7

4645975

6.54

663710

South Dakota

1

819761

1.15

819761

Tennessee

9

6375431

8.97

708381

Texas

36

25268418

35.55

701900

Utah

4

2770765

3.90

692691

Vermont

1

630337

0.89

630337

Virginia

11

8037736

11.31

730703

Washington

10

6753369

9.50

675336

West Virginia

3

1859815

2.62

619938

Wisconsin

8

5698230

8.02

712278

Wyoming

1

568300

0.80

568300

The Mathematics

The algorithm finally settled on by Congress was chosen because it was thought to be the fairest. There are different ways of
defining what "fair" means, and so it cannot be settled mathematically.
In this context "fair" is taken to mean "minimizes the relative difference in representatives per person between states".

To see how the algorithm meets this definition of fairness, let's see what happens when we examine any two states to see if
transferring one seat between them would improve the apportionment. This is the argument published by E. V. Huntington in [4].

Suppose after the apportionment, state \(A\) has received \(x+1\) seats, and state \(B\) has received \(y\) seats.
Furthermore, also suppose that \(A\) is over-represented because the number of people per representative is less than for \(B\):

Which is true. (The numbers also tally with the U.S. Census Bureau [6], and my program to calculate apportionments [5], where the priority value for California's last seat is \(711,308\), which is \(37,341,989/\sqrt{52 \times 53}\).)

and the relative difference is smaller before the seat transfer (using (\ref{Adiff}) and (\ref{Bdiff})). So the original apportionment is optimal.
There was nothing special about the choice of \(A\) and \(B\), so we can conclude that the apportionment is optimal overall.

Again, this checks out for our example. The relative difference for 53 seats for California and 27 for New York is \(0.021\), versus \(0.035\) for 52 for California and 28 for New York.